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J. Phys. Chem. A 2007, 111, 45224526
Density Functional Theory Treatment of Electron Correlation in the NuclearElectronic Orbital Approach Michael V. Pak, Arindam Chakraborty, and Sharon HammesSchiffer* Department of Chemistry, 104 Chemistry Building, PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 ReceiVed: January 18, 2007; In Final Form: March 7, 2007
This paper presents the nuclearelectronic orbital density functional theory [NEODFT(ee)] method for including electronelectron correlation and nuclear quantum effects selfconsistently in quantum chemical calculations. The NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically. In the NEODFT(ee) approach, the correlated electron density is used to obtain the nuclear molecular orbitals, and the resulting nuclear density is used to obtain the correlated electron density during an iterative procedure that continues until convergence of both the nuclear and electronic densities. This approach includes feedback between the correlated electron density and the nuclear wavefunction. The application of this approach to bihalides and acetylene indicates that the nuclear quantum effects do not significantly impact the electron correlation energy, but the quantum nuclear energy is enhanced in the NEODFT(ee) B3LYP method. The excellent agreement of the NEODFT(ee)optimized bihalide structures with the vibrationally averaged geometries from gridbased quantum dynamical methods provides validation for the NEODFT(ee) approach. Electronproton correlation could be included by the development of an electronnucleus correlation functional. Alternatively, explicit electronproton correlation could be included directly into the NEO selfconsistentfield framework with Gaussiantype geminal functions.
I. Introduction Conventional electronic structure calculations invoke the BornOppenheimer approximation, where the nuclei are represented as classical point charges. Nuclear quantum effects such as zero point motion and hydrogen tunneling, however, have been shown to be significant for systems involving hydrogen bonding interactions and hydrogen transfer reactions. Recently, methods have been developed to avoid the BornOppenheimer approximation and include nuclear quantum effects in electronic structure calculations.116 In particular, the nuclearelectronic orbital (NEO) approach17 is designed to treat selected nuclei (e.g., the hydrogen nuclei involved in hydrogen bonding interactions or hydrogen transfer reactions) quantum mechanically, while the remaining nuclei are treated classically. In the NEO approach, mixed nuclearelectronic wavefunctions are calculated variationally using molecular orbital techniques with Gaussian basis functions for the electrons and the nuclei. In conventional electronic structure theory, electron correlation is defined as the difference between the exact energy and the singleconfiguration HartreeFock energy. Nondynamical electron correlation arises from the influence of other configurations that are close in energy to and mix strongly with the HartreeFock configuration, and dynamical electron correlation arises from interactions of electrons at close separation distances, which are difficult to describe. Typically, multiconfigurational methods are used to include nondynamical electron correlation, and methods such as perturbation theory, coupled cluster theory, configuration interaction, and density functional theory (DFT) are used to include dynamical electron correlation. * To whom correspondence should be addressed. Email: shs@ chem.psu.edu.
Within the framework of the NEO approach, both electronelectron and electronproton correlation effects play important roles. We have shown that electronproton dynamical correlation strongly impacts the calculation of nuclearelectronic wavefunctions because of the attractive electrostatic interaction between the electron and the proton.2,3 Nondynamical electronelectron and electronproton correlation effects have been included in the NEO framework with the configuration interaction (NEOCI), multiconfigurational selfconsistentfield (NEOMCSCF), and nonorthogonal configuration interaction (NEONOCI) methods.14 Dynamical electronelectron and electronproton correlation effects have been included with secondorder perturbation theory (NEOMP2).5 More recently, an explicitly correlated HartreeFock (NEOXCHF) approach was developed to include explicit electronproton correlation directly into the NEO selfconsistentfield framework using Gaussiantype geminal functions.7 In contrast to previous approaches, which produce nuclear wavefunctions that are too localized and that severely overestimate hydrogen vibrational frequencies,1220 the NEOXCHF method provides accurate nuclear wavefunctions and hydrogen vibrational stretch frequencies.7 The NEOXCHF method is computationally practical for manyelectron systems with a relatively small number of quantum nuclei because only electronproton correlation is treated explicitly. A more computationally tractable method must be used to include electronelectron correlation for these types of systems. In this paper, we present the NEODFT(ee) approach for including electronelectron correlation within the NEO framework in a selfconsistent and computationally practical manner. The theoretical foundation for nonBornOppenheimer density functional theory was provided by Parr and coworkers.21 More recently, Shigeta and coworkers presented a theoretical for
10.1021/jp0704463 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/19/2007
DFT of Electron Correlation in the NEO Approach
J. Phys. Chem. A, Vol. 111, No. 20, 2007 4523
mulation and numerical scheme for nonBornOppenheimer DFT calculations.8 In addition, Kreibich and Gross developed a multicomponent density functional theory for a combined system of electrons and nuclei and provided approximate functionals for the electronnuclear correlation energy.14 These previous methods were developed for systems in which all of the nuclei and all of the electrons were treated quantum mechanically, and the electronproton correlation was included via an electronnuclear functional. In contrast, the NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically.22 Moreover, an alternative to developing an electronnuclear functional is to include only electronelectron correlation with density functional theory and to include electronproton correlation explicitly with the NEOXCHF approach. The paper is organized as follows. Section II presents the the oretical formulation and implementation of the NEODFT(ee) method. Section III describes the application of this approach to bihalides and acetylene for benchmarking purposes. The electron correlation effects and the nuclear quantum effects are analyzed, and the bihalide structures are compared to results from gridbased quantum dynamical methods and to experimental data. Concluding remarks are provided in Section IV. II. Theory and Methods A. Theoretical Formulation and Implementation of NEODFT(ee). The NEO approach is formulated for a system with Ne electrons, Np quantum nuclei, and Nc classical nuclei. This approach is designed for systems in which at least two nuclei are treated classically, thereby eliminating the difficulties associated with translations and rotations.23 For simplicity, here we assume that the quantum nuclei are protons with a charge of +1. The equations are easily extended for other types of quantum nuclei. In this case, the NEOHF (HartreeFock) Hamiltonian is defined as
H ˆ tot ) Ne
implemented. For this case, the electronic and nuclear Fock equations are
f e(1)ψei (1) ) (he(1) + νee(1) + νep,e(1))ψei (1) ) ei ψei (1) (4) f p(1′)ψi′p(1′) ) (hp(1′) + νpp(1′) + νep,p(1′))ψi′p(1′) ) i′p ψi′p(1′) (5) The two particle operators are defined as
νee(1)ψej (1) ) 2
e e e 1 e e ∑b 〈ψeb(2)r1 12 ψb(2)〉ψj (1)  ∑ 〈ψb(2)r12 ψj (2)〉ψb(1) b
(6) νpp(1′)ψj′p(1′) )
p 1 p p 1 p 〈ψb′ (2′)r1′2′ ψb′ (2′)〉ψj′p(1′)  ∑ 〈ψb′ (2′)r1′2′ ψj′p(2′)〉ψb′ (1′) ∑ b′ b′
(7)
νep,e(1)ψej (1) ) 
p 1 p 〈ψb′ (1′)r11′ ψb′ (1′)〉ψej (1) ∑ b′
νep,p(1′)ψjp′(1′) ) 2
1 ψeb(1)〉ψjp′(1′) ∑b 〈ψeb(1)r11′
(8) (9)
where the symbol ψ denotes the spatial orbitals, and the summations of b and b′ are over the occupied electronic and proton spatial orbitals, respectively. Here, νee(i) and νpp(i′) are the Coulombexchange operators for the electrons and protons, respectively, and νep,e(i) and νep,p(i′) are the electronproton Coulomb operators for the electrons and protons, respectively. The electronproton Coulomb operator for the proton can be expressed in terms of the total electron density
Fe(r) ) 2
∑b ψeb(r)2
where Np
Np
Np
∑i he(i) + ∑i ′ hp(i′) + ∑i ′ j∑ ′>i ′ r
1
Ne
+
i ′j ′
Ne
1
∑i ∑ j>i r
ij
Ne

Np
1
∑i ∑i ′ r
ii ′
(1) where the unprimed indices i, j refer to electrons and the primed indices i′, j′ refer to the quantum protons. he and hp are the oneparticle terms for the electrons and protons, respectively, and are defined as
1 he(i) )  ∇2i 2
Nc
ZA
∑A r
1 2 ∇i ′ + hp(i′) ) 2M
(2)
iA
Nc
ZA
∑A r
(3) i ′A
where the mass of the proton is denoted by M, the index A refers to classical nuclei, and the charges of the classical nuclei are represented by ZA. We have derived the NEOHF equations for a restricted HartreeFock (RHF) treatment of the electrons and a highspin openshell treatment of the quantum nuclei. The extension to other spin configurations is straightforward and has been
νep,p(r1′) ) 
∫
Fe(r1) dr1 r11′
(10)
In the NEODFT(ee) approach, the total electron density Fe(r) is obtained from a conventional electronic structure DFT calculation, thereby reflecting the changes in the electronic environment due to electronelectron correlation. The modified electronproton Coulomb operator in eq 10 is used in eq 5 for the solution of the nuclear Fock equations to obtain the nuclear molecular orbitals for the quantum protons. Conventional electronic structure DFT calculations are based on the KohnSham orbital equations24,25
(21 ∇ + ν (1))ψ (1) ) ψ (1) 2 1
e eff
e i
e i
e i
(11)
where the effective potential νeeff(r) is Nc
νeeff(r1) ) 
ZA
∑A r
1A
+
∫
Fe(r2) r12
dr2 + νexc(r1)
(12)
and νexc(r) is the exchangecorrelation potential. In the NEODFT(ee) approach, the contribution to the effective potential associated with the external potential created by the classical
4524 J. Phys. Chem. A, Vol. 111, No. 20, 2007
Pak et al.
nuclei is modified to account for the quantum treatment of selected nuclei Nc
νeeff (r1)
)
ZA
∑A r
Np

1A
∫ ∑ b′
p ψb′ (r1′)2
∫
r11′ Fe(r2)
dr1′ +
dr2 + νexc(r1)
r12
(13)
The modified effective potential in eq 13 is used in eq 11 to calculate the total electron density. The NEODFT(ee) approach requires the iterative solution of the nuclear NEOHF equations (eq 5) and the conventional electronic structure DFT equations (eq 11) until convergence of both the nuclear and electronic densities. The NEOHF equations for the quantum nuclei depend on the total electron density and provide the nuclear molecular orbitals. The expression for the nuclear HF energy is Np
EHF p
∑ b′
)
[
1 p p p p p 〈ψb′ h ψb′ 〉 + 〈ψb′ νppψb′ 〉2
∫∫
p Fe(r1)ψb′ (r1′)2
dr1′ dr1
r11′
]
(14)
The conventional electronic structure DFT equations depend on the nuclear molecular orbitals and provide the total electron density. The expression for the electronic DFT energy is Ne/2
EDFT e
)2
∑b
〈 〉 ψeb
1 1  ∇2 ψeb + 2 2
∫∫
Fe(r1)ZA
A
r1A
∫Vxc(r1)F (r1)dr1  ∑ ∫ ∫∫ ∑ b′
r12
Nc
e
Np
Fe(r1)Fe(r2)
dr1dr2 +
dr1 
p Fe(r1)ψb′ (r1′)2
r11′
dr1′ dr1
(15)
After convergence of the nuclear and electronic densities, the total energy is calculated as Np
E)
EHF p
+
EDFT e
+
∫∫ ∑ b′
p Fe(r1)ψb′ (r1′)2
r11′
dr1′ dr1
TABLE 1: Electron Correlation Energies and Quantum Nuclear Energies for the (FHF) Moleculea method
Ecorr (classical H)
Ecorr (quantum H)
∆EQN
HF DFT BLYP DFT B3LYP MP2 CCSD(T)
0.0000000 0.734705 0.804013 0.578704 0.591654
0.000000 0.734783 0.800583 0.577176 0.590109
0.032118 0.032040 0.035547 0.033645 0.033663
a Ecorr is defined as the difference between the HF energy and the energy obtained with the specified electronic structure method. ∆EQN is defined as the difference between the energies obtained with a quantum H and a classical H for the specified electronic structure method. Energies are given in atomic units.
with the vibrational selfconsistentfield (VSCF) method.27 At the VSCF level, the total molecular vibrational wavefunction is constructed as a product of singlemode wavefunctions. The singlemode wavefunctions are generated on a grid and are optimized using a selfconsistentfield procedure. The expectation values of the Cartesian coordinates with respect to the VSCF groundstate wavefunction are determined through linear transformations. For the VSCF calculations presented below, we used a locally modified version of GAMESS that includes the option to calculate vibrationally averaged molecular structures within the VSCF framework.19,26 For these calculations, 16 directly computed quadrature points along each normal mode were used to construct the VSCF potential. We also performed twodimensional (2D) vibrational configuration interaction (VCI) calculations that did not include the bending modes of the bihalides. In the 2D VCI method,2830 the twodimensional vibrational wavefunction is represented as a linear combination of twodimensional harmonic oscillator wavefunctions, and the expansion coefficients are obtained variationally by diagonalizing the Hamiltonian matrix in the twodimensional harmonic oscillator basis. These calculations were carried out using seven harmonic oscillator basis functions per mode and an elevenpoint GaussHermite quadrature scheme for the numerical integration. Convergence with respect to the number of basis functions and quadrature points was investigated by repeating the calculations with five harmonic oscillator basis functions and seven GaussHermite quadrature points per mode. The vibrationally averaged distances differed by only 0.001 Å for these two sets of calculations, thereby confirming convergence. III. Results and Discussion
(16)
In this expression for the total energy, the electronproton interaction term is subtracted from the sum of the nuclear HF and the electronic DFT energies to avoid double counting of this interaction, which is included in both the nuclear HF and the electronic DFT energies. Since only the external potential part of the DFT effective potential is effected by these modifications, this procedure can be applied in conjunction with any kind of DFT functional (i.e., purely exchange or mixed exchangecorrelation functionals). This general procedure is closely related to the formulation proposed by Parr and coworkers21 for the quantum mechanical treatment of all nuclei. The NEODFT(ee) method, however, is designed to treat only a relatively small number of nuclei quantum mechanically and the remaining nuclei classically. The NEODFT(ee) method has been incorporated into the GAMESS electronic structure code.26 B. GridBased Methods. For benchmarking purposes, we calculated vibrationally averaged geometries for the bihalides
In this section, we present calculations for the bihalides (FHF) and (ClHCl) and for acetylene. The aug′ccpVTZ electronic basis set,3133 in which the prime indicates that diffuse functions were placed only on the heavy atoms, was used for the bihalide calculations, and the 6311G electronic basis set34 was used for the acetylene calculations. For the NEO calculations, all hydrogen nuclei were treated quantum mechanically with the DZSPDN nuclear basis set.1 We calculated both the electron correlation energies and the quantum nuclear energies for all three molecules with secondorder perturbation theory (MP2),35 coupled cluster theory [CCSD(T)],36,37 and DFT with the BLYP38 and B3LYP3941 functionals. These calculations were performed at the equilibrium HartreeFock geometries. The electron correlation energy Ecorr is defined to be the difference between the HartreeFock (HF) energy and the energy obtained with the specified electronic structure method. When the hydrogen nuclei are treated classically, the electron correlation energies are determined relative to the conventional HF energy, and when the
DFT of Electron Correlation in the NEO Approach
J. Phys. Chem. A, Vol. 111, No. 20, 2007 4525
TABLE 2: Electron Correlation Energies and Quantum Nuclear Energies for the (ClHCl) Moleculea
TABLE 4: Experimental and Theoretical XX Distances for the Bihalide Systems (FHF) and (ClHCl)a
method
Ecorr (classical H)
Ecorr (quantum H)
∆EQN
method
(FHF) RFF
(ClHCl) RClCl
HF DFT BLYP DFT B3LYP MP2 CCSD(T)
0.000000 1.350802 1.460847 0.429777 0.479245
0.000000 1.344543 1.460767 0.428383 0.477675
0.030822 0.030901 0.037080 0.032216 0.032393
experimental Re experimental R0 MP2e 4D VSCFMP2e 2D gridMP2f 2D VCIMP2 NEOMP2(ee)e DFT 4D VSCFDFT 2D VCIDFT NEODFT(ee)
2.277 71b 2.304d 2.288 2.311 2.324 2.324 2.321 2.301 2.325 2.336 2.329
3.1122(26)c 3.14676(5)c 3.110 3.131 3.152 3.152 3.150 3.153 3.175 3.196 3.184
a Ecorr is defined as the difference between the HF energy and the energy obtained with the specified electronic structure method. ∆EQN is defined as the difference between the energies obtained with a quantum H and a classical H for the specified electronic structure method. Energies are given in atomic units.
TABLE 3: Electron Correlation Energies and Quantum Nuclear Energies for the Acetylene Moleculea method
Ecorr (classical H)
Ecorr (quantum H)
∆EQN
HF DFT BLYP DFT B3LYP MP2 CCSD(T)
0.000000 0.642525 0.723615 0.254133 0.270923
0.000000 0.642723 0.718516 0.254782 0.270701
0.065776 0.065578 0.070875 0.065127 0.065997
a Ecorr is defined as the difference between the HF energy and the energy obtained with the specified electronic structure method. ∆EQN is defined as the difference between the energies obtained with a quantum H and a classical H for the specified electronic structure method. Energies are given in atomic units.
hydrogen nuclei are treated quantum mechanically, the electron correlation energies are determined relative to the NEOHF energy. The quantum nuclear energy ∆EQN is defined to be the difference between the energies obtained with quantum hydrogen nuclei and classical hydrogen nuclei for the specified electronic structure method. The electron correlation effects are treated in a different manner for the MP2, CCSD(T), and DFT methods. In both the NEOMP2 and NEOCCSD(T) approaches, initially the nuclear and electronic molecular orbitals are obtained from a NEOHF calculation, and subsequently the electron correlation corrections are evaluated with either the MP2 or the CCSD(T) method. Thus, the impact of electron correlation on the nuclear wavefunction is not included in these approaches. In the NEODFT approach, however, the correlated electron density is used to obtain the nuclear molecular orbitals at each step of the iterative procedure, and the resulting nuclear density is used to calculate the correlated electron density for the subsequent step of this procedure. Thus, the NEODFT approach includes feedback between the correlated electron density and the nuclear wavefunction. The results for the (FHF), (ClHCl), and acetylene molecules are given in Tables 1, 2, and 3, respectively. These tables illustrate several qualitative trends for all three systems. The DFT method appears to recover more electron correlation energy than the MP2 and CCSD(T) methods. More importantly, these results show that the electron correlation energy is similar for the conventional and the NEO calculations at each level of electronic structure theory. This observation indicates that the quantum treatment of the hydrogen nuclei does not significantly impact the electron correlation energy for the MP2, CCSD(T), and DFT methods. The results also show that the quantum nuclear energies are similar for the MP2, CCSD(T), and DFT methods. This observation indicates that the impact of the quantum treatment of the hydrogen nuclei on the energy is relatively independent of the level of electronic structure theory. For all three molecules, however, the quantum nuclear energies
a The aug′ccpVTZ electronic basis set was used in all calculations. The DZSPDN nuclear basis set was used in all of the NEO calculations. The DFT calculations use the B3LYP functional. Distances are given in angstroms. b Ref 44. c Ref 45. d Calculated using the rigid rotor approximation and B0 from ref 44. e Ref 19. f Ref 42.
are greatest at the NEODFT(ee) B3LYP level. This observation suggests that the quantum nuclear energy is enhanced by the combination of the larger electron correlation energy and the feedback between the correlated electron density and the nuclear wavefunction included in the NEODFT(ee) B3LYP approach. In order to benchmark the NEODFT(ee) method, we compare the FF and ClCl distances for the bihalides optimized with the NEO approach to vibrationally averaged geometries obtained with the fulldimensional VSCF method and the 2D VCI method. The optimized NEO XX distances correspond to the minimum energy geometry at the specified level of electronic structure theory within the NEO framework. The results for these calculations at both the MP2 and DFT B3LYP levels of electronic structure theory are given in Table 4. The agreement between the 2D VCIMP2 and the 2D gridMP2 calculations from Del Bene and Jordan42 to within 0.001 Å provides validation for the VCI results. We emphasize that the NEO calculations are much faster than the fulldimensional VSCF and 2D VCI calculations. For the (FHF) system, the NEODFT(ee) optimization required ∼10 min of CPU time, while the fulldimensional VSCF and 2D VCI calculations required ∼35 and ∼2 h of CPU time, respectively. As discussed previously,19 the XX distances are shorter for the fulldimensional VSCF calculations than for the 2D VCI calculations because the inclusion of the bending motions tends to decrease the XX distance. The NEO approach is capable of including the nuclear quantum effects of the bending mode with an adequate nuclear basis set and sufficient electronproton correlation, but the present implementation is directed at an accurate description of the stretching modes and is not optimized for the description of the lowerfrequency bending modes. As a result, the NEOMP2(ee) XX distances are within 0.003 Å of the 2D VCIMP2 distances and are greater than the fulldimensional VSCFMP2 distances by ∼0.010.02 Å. Similar trends are observed for the NEODFT(ee) XX distances, which are within ∼0.01 Å of the 2D VCIDFT distances. The slightly larger differences between the NEO and 2D VCI calculations observed for DFT may arise from the feedback between the correlated electron density and the nuclear molecular orbitals in the NEODFT(ee) approach. This feedback is not present in the 2D VCI and NEOMP2(ee) methods, and it may lead to a better description of the bending mode. Table 4 indicates that the NEODFT(ee) XX distances are slightly closer to the fulldimensional VSCFDFT distances than to the 2D VCIDFT distances. This better agreement with the fulldimensional gridbased calculations also suggests that the NEO
4526 J. Phys. Chem. A, Vol. 111, No. 20, 2007 DFT(ee) approach describes the bending mode more accurately than the NEOMP2(ee) method. Note that the DFT distances do not agree as well as the MP2 distances with the experimental distances. The larger bond lengths for DFT than for MP2 are a characteristic of the DFT B3LYP method for these molecules with this electronic basis set and are not a reflection of the accuracy of the NEO methodology. Similar trends were observed in previous conventional electronic structure studies.43 For benchmarking purposes, our goal is to compare the NEODFT(ee) method to gridbased quantum dynamical methods at the same level of electronic structure theory. The excellent agreement between the distances obtained with the NEODFT(ee) method and the 2D VCIDFT and fulldimensional VSCFDFT methods provides validation for the NEODFT(ee) approach. IV. Concluding Remarks In this paper, we presented the NEODFT(ee) approach for including both electronelectron correlation and nuclear quantum effects selfconsistently within the NEO framework. The NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically. In the NEODFT(ee) approach, the correlated electron density is used to obtain the nuclear molecular orbitals, and the resulting nuclear density is used to obtain the correlated electron density during an iterative procedure that continues until convergence of both the nuclear and electronic densities. As a result, this approach includes feedback between the correlated electron density and the nuclear wavefunction. Our analysis of NEO and conventional electronic structure calculations at various levels of theory for bihalides and acetylene indicates that the nuclear quantum effects do not significantly impact the electron correlation energy. This analysis also suggests that the quantum nuclear energy is enhanced in the NEODFT(ee) B3LYP method due to the combination of greater electron correlation energy and feedback between the correlated electron density and the nuclear wavefunction. Benchmarking calculations on the bihalides by comparison of the NEODFT(ee) method to gridbased quantum dynamical methods provide an assessment of the accuracy of this approach. The current version of the NEODFT(ee) method includes only electronelectron correlation with density functional theory. Electronproton correlation plays a vital role in the description of nuclearelectronic wavefunctions because of the attractive electronproton electrostatic interaction. Inadequate treatment of electronproton correlation produces nuclear wavefunctions that are too localized and leads to severe overestimations of hydrogen vibrational frequencies. One potential direction for the inclusion of electronproton correlation is to develop an electronnucleus correlation functional within the density functional theory framework. Alternatively, explicit electronproton correlation can be included directly into the NEO selfconsistentfield framework with Gaussiantype geminal functions using the NEOXCHF method.7 The NEOXCHF method has already been shown to provide accurate descriptions of nuclearelectronic wavefunctions. The combination of the NEODFT(ee) approach for electronelectron correlation and the NEOXCHF approach for electronproton correlation would provide a computationally practical and highly accurate method for inclusion of nuclear quantum effects in electronic structure calculations. Acknowledgment. We gratefully acknowledge the support of AFOSR Grant No. FA95500710143 and helpful discussions with Simon Webb and Chet Swalina.
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